Where the Conflict Really Lies: Science, Religion, and Naturalism (37 page)

Of course it is always possible to maintain that these mathematical powers are a sort of spandrel, of no adaptive use in themselves, but an inevitable accompaniment of other powers that do promote reproductive fitness. The ability to see that 7 gazelles will provide much more meat than 2 gazelles is of indisputable adaptive utility; one could argue that these more advanced cognitive powers are inevitably connected with that elementary ability, in such a way that you can’t have the one without having the other.

Well, perhaps; but it sounds pretty flimsy, and the easy and universal availability of such explanations makes them wholly implausible. It’s like giving an evolutionary explanation of the music of Mozart and Bach in terms of the adaptiveness, the usefulness, in the Pleistocene, of rhythmical movement in walking or running long distances.
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There is a third way in which the “unreasonable efficacy” of mathematics in science points to and exemplifies deep concord between theistic religion and science. Mathematics, naturally enough, is centrally about numbers and sets. But numbers and sets themselves make a great deal more sense from the point of
view of theism than from that of naturalism. Now there are two quite different but widely shared intuitions about the nature of numbers and sets. First, we think of numbers and sets as abstract objects, the same sort of thing as propositions, properties, states of affairs and the like. It natural to think of these things as existing necessarily, such that they would have been there no matter how things had turned out. (After all, we think of some propositions—true mathematical propositions, for example—as necessarily true; but a proposition can’t be necessarily true without existing necessarily.) On the other hand, there is another equally widely shared intuition about these things: most people who have thought about the question, think it incredible that these abstract objects should just exist, just
be
there, whether or not they are ever thought of by anyone. Platonism with respect to these objects is the position that they do exist in that way, that is, in such a way as to be independent of mind; even if there were no minds at all, they would still exist. But there have been very few real Platonists, perhaps none besides Plato and Frege, if indeed Plato and Frege were real Platonists (and even Frege, that alleged arch-Platonist, referred to propositions as
gedanken
, thoughts). It is therefore extremely tempting to think of abstract objects as ontologically dependent upon mental or intellectual activity in such a way that either they just are thoughts, or else at any rate couldn’t exist if not thought of. (According to the idealistic tradition beginning with Kant, propositions are essentially
judgments
.)

But if it is
human
thinkers that are at issue, then there are far too many abstract objects. There are far too many real numbers for each to have been thought of by some human being. The same goes for propositions; there are at least as many propositions as there are real numbers. (For every real number
r
, for example, there is the proposition that
r
is distinct from the Taj Mahal.) On the other hand, if abstract objects were divine thoughts, there would be no problem here. So perhaps the most natural way to think about abstract objects, including numbers, is as divine thoughts.
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Second, consider sets. Perhaps the most common way to think of sets is as displaying at least the following characteristics: (1) no set is a member of itself; (2) sets (unlike properties) have their extensions essentially; hence sets with contingently existing members are themselves contingent beings, and no set could have existed if one of its members had not; (3) sets form an iterated structure: at the first level, there are sets whose members are nonsets, at the second, sets whose members are nonsets or first level sets, and so on.
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(Note that on this iterative conception, the elements of a set are in an important sense prior to the set. That is why on this conception no set is a member of itself, thus disarming the Russell paradoxes in their set theoretical form.
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)

It is also natural to think of sets as
collections
—that is, things whose existence depends upon a certain sort of intellectual activity—a collecting or “thinking together.” Thus Georg Cantor: “By a ‘set’ we understand any collection M into a whole of definite, well-distinguished objects of our intuition or our thought (which will be called the ‘elements’ of M).”
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According to Hao Wang, “the set is a single object formed by collecting the members together.”
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And

It is a basic feature of reality that there are many things. When a multitude of given objects can be collected together, we arrive at a set. For example, there are two tables in this room. We are ready to view them as given both separately and as a unity, and justify this by pointing to them or looking at them or thinking about them either one after the other or simultaneously. Somehow the viewing of certain given
objects together suggests a loose link which ties the objects together in our intuition.
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If sets
were
collections, that would explain their having the first three features. (First, if sets were collections, the result of a collecting activity, the elements collected would have to be present before the collecting; hence no set is a member of itself. Second, a collection could not have existed but been a collection of items different from the ones actually collected, and a collection can’t exist unless the elements collected exist; hence collections have their members essentially, and can’t exist unless those members do. And third, clearly there are non-collections, then first level collections whose only members are noncollections, then second level collections whose members are noncollections or first level collections, et cetera.) But of course there are far too many sets for them to be a product of
human
thinking together. Furthermore, many sets are such that no human being could possibly think all their members together—for example, the set of real numbers. Therefore there are many sets such that no human being has ever thought their members together, many such that their members have not been thought together by any human being. That requires an infinite mind—one like God’s.

The basic objects of mathematics, that is, numbers and sets, fit very neatly into a theistic way of looking at the world—vastly better than into a naturalistic perspective. Perhaps this explains the strenuous efforts, on the part of Hartry Field and others, to “reinterpret” mathematics in such a way as to make it possible for naturalism to accommodate it.
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Again, we see deep concord between theistic religion and science.

There is still another way in which theism is friendly to mathematics, more friendly than naturalism is. The objects of mathematics—numbers, functions, sets—are
abstract
objects. Abstract objects, so
we think, differ from concrete objects in that they do not occupy space and do not enter into causal relations. The number 3 can’t cause anything to happen; it is causally inert. This is not a peculiarity of that number; the same goes for all the other numbers—real, complex, whatever—and for sets, including functions. But this creates a puzzle.
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It seems sensible to think that the objects we can know about can causally affect us in some way, or at least stand in causal relationship with us. We know about trees. We can perceive them; this involves light waves being reflected from trees into our eyes, forming an image on the retina; this induces electrical activity in the optic nerve, finally issuing in neural activity in the brain. We know something about distant galaxies, again, only because electromagnetic radiation from them reaches us. As I say, it seems sensible to think that a necessary condition of our knowing about an object or kind of object is our standing in some kind of causal relation to that object or kind of object. If this is so, however, and if, furthermore, numbers and their kin are abstract objects, then it looks as though we couldn’t know anything about them.

Once again, theism is relevant. According to classical versions of theism, sets, numbers and the like, as I argued above, are best conceived as divine thoughts. But then they stand to God in the relation in which a thought stands to a thinker. This is presumably a
productive
relation: the thinker produces his thoughts. It is therefore also a causal relation. If so, then numbers and other abstract objects also stand in a causal relation to us. For we too stand in a causal relation to God; but then anything else that stands in a causal relation to God stands in a causal relation to us. Therefore numbers and sets stand in a causal relation to us, and the problem about our knowing these things disappears.

Another and perhaps less obvious condition for the success of science has to do with our ways of learning from experience. We human beings take it utterly for granted that the future will resemble the past. As David Hume pointed out with his usual keen insight, in the past we have found bread but not stones to be nourishing (this may have been known even before Hume); we expect the former to continue to have this salubrious property and the latter to lack it. Past ax heads dropped into water have sunk; we expect the next one to do the same. Night has always followed day: we assume in consequence that today will be followed by tonight. It is only by virtue of this assumption, furthermore, that we are able to learn from experience. Of course we don’t expect the future to resemble the past in
every
respect; I have no doubt, for example, that my grandchildren will be larger ten years from now. Saying precisely how we expect the future to resemble the past is no mean task; we expect the future to resemble the past in relevant respects; but specifying the relevant respects is far from easy. Nevertheless, we do expect the future to resemble the past, and this expectation is crucial to our being able to learn from experience.

We generalize what we learn to the future; but this is not the extent of our generalization of experience. Aristotle held that heavy objects fall faster than light objects; according to scientific folklore, Galileo dropped a couple of balls of unequal weights from the leaning Tower of Pisa, noted that they fell at the same rate, and concluded that Aristotle was wrong. We run a few experiments, and conclude that Newton’s law of gravity is at least approximately true. In cases like these we don’t conclude merely that Aristotle’s theory is false for that pair of balls Galileo allegedly dropped, or for the area around the leaning tower, or on Thursdays. We don’t conclude that maybe Aristotle was right in his day—two thousand years ago, maybe heavy
things did fall faster than light. No; we conclude that Aristotle’s theory is false generally, and that Galileo’s results hold for any pair of balls that might be dropped. In experiments verifying Newton’s laws, we don’t infer merely that Newton’s laws held in the time and place where those experiments were conducted; we think they hold much more generally. We don’t necessarily conclude that they hold for all of time and space (we are open to the idea that things may have been different shortly after the big bang, or in one of those other universes of which cosmologists speak); but we do conclude that they hold far beyond the temporal and spatial limits of the situation of the experiments. The great eighteenth-century philosopher Thomas Reid claimed that among the “principles of contingent truth” is that “
in the phaenomena of nature, what is to be, will probably be like to what has been in similar circumstances
” (Reid’s emphasis). What he meant is that we simply find ourselves, by virtue of our nature, making that assumption. This principle, furthermore, “is necessary for us before we are able to discover it by reasoning, and therefore is made a part of our constitution, and produces its effects before the use of reason.”
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Reid goes on to claim that having this conviction—that “in the phaenomena of nature, what is to be, will probably be like to what has been in similar circumstances”—is essential to learning from experience. This isn’t exactly right, at any rate if what Reid means is that one must explicitly have this belief in order to learn from experience. A child learns from experience; “the burnt child dreads the fire.” The burnt child may never have raised the question whether the future
resembles the past, and she may have no explicit views at all on that topic. What learning from experience requires is more like a certain habit, a certain practice—the habit of making inductive inferences. But that too isn’t exactly right: in any event there need be nothing like explicitly thinking of premise and conclusion. It is more as if we have the experience and in direct response to it form a belief that goes far beyond the confines of the experience.

We are able to learn that unsupported rocks near the surface of the earth will fall down rather than up, that water is good to drink, that rockfall is dangerous—we learn these things only by virtue of exercising this habit. Indeed, it is only by virtue of this habit that a child is able to learn a language. (My parents teach me “red”; I get the idea and see what property they express by that word; unless I proceed in accord with this habit, I shall have to start over the next time they use “red.”) Our ordinary cognitive life deeply depends on our making this assumption, or following this practice.

Of course this holds for the practice of science as well as for everyday cognitive life. According to the story noted above, Galileo dropped two balls, one heavy and one light, off the leaning Tower of Pisa, to see if Aristotle was right in thinking heavy objects fall faster than light. Aristotle was wrong; they fell at the same rate. Presumably no one suggested that Galileo should perhaps perform this experiment every day, on the grounds that all he had shown was that on that particular day heavy and light objects fall at the same rate. No one suggested that the experiment should be repeated in Asia, or that we should look for other evidence on the question whether in Aristotle’s time he may have been right. A crucial experiment may be repeated, but not because we wonder whether the same circumstances will yield the same result.